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I know that in LA, you can represent a square matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ with its eigenvalues and eigenvectors.

Why is it not possible to have an equivalent equation for infinite dimension? Like for a self-adjoint compact linear operator in Hilbert space, can it somehow be represented with its eigenfunctions/eigenvalues?

Am I getting mixed up? I just started functional analysis very recently - so any extra explanation would truly be appreciated.

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    There are such theorems, you will encounter them even in a basic course of functional analysis, but may be they are not as general as they are in finite dimensions (you get good results in special cases). I haven't taken a formal course in functional analysis (I'm only in seventh) but course material online usually covers a list of topics, and I read up the material from the textbook (in my case, Rudin). – 2017-01-26
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    The result you referred to is known as the spectral theorem, for the compact self-adjoint case one has that the spectrum consists of eigenvalues and only accumulates at $0$, with an orthonormal basis of eigenvectors(as analogous to the finite dimensional case). The most general version I know of is for closed self-adjoint operators, which is stated using multiplication operators. – 2017-01-26

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